## 1. Introduction

It is well established that gravity waves (GWs) have a substantial influence on the large-scale atmospheric circulation, particularly in the middle atmosphere. As a result, sources of atmospheric GWs have received a great deal of attention. Significant tropospheric sources include topography, convective and frontal activities (Bretherton and Smolarkiewicz 1989; Shutts and Gray 1994), wind shear instabilities (Lalas and Einaudi 1976; Rosenthal and Lindzen 1983; Lott et al. 1992), nonmodal growth (Lott 1997; Bakas and Ioannou 2007), and geostrophic adjustment.

When it comes to geostrophic adjustment, one should distinguish classical adjustment, in which an initially unbalanced flow radiates GWs as it returns to near-geostrophic balance (Rossby 1937, Blumen 1972, Fritts and Luo 1992), from spontaneous adjustment, in which a well-balanced flow radiates weakly GWs in the course of its (near balanced) evolution (Ford et al. 2000). The source of GW activity differs between the two types of adjustment. In the first case, GW generation should be attributed to the mechanism responsible for the initial imbalance rather than to the adjustment.^{1} In the second case, spontaneous adjustment itself is the GW source.

Spontaneous adjustment, on which the present paper focuses, has been the subject of intense research activity. Much of this has been motivated by theoretical issues related to the limitations of balanced models and the nonexistence of exactly invariant slow manifolds (e.g., Vanneste 2008 and references therein). There are, however, practical implications, in particular for the parameterization of nonorographic GWs in general circulation models, since the (pseudo) momentum flux associated with the GWs generated spontaneously may well be significant. To assess this, it is important to quantify the GW activity generated by simple, physically plausible processes. This is the main aim of this paper.

The process that we examine is the generation of GWs that results from the advection of potential vorticity (PV) anomalies by a vertically sheared wind. Since we consider background flow with uniform PV, it is related to the work of Plougonven et al. (2005) on the unbalanced instabilities associated with surface edge waves (see also Molemaker et al. 2005). In our case, however, there is no boundary, and the PV disturbance is imposed within the flow and can have a finite vertical extent. The process that we examine is also related to the work of Vanneste and Yavneh (2004) and Olafsdottir et al. (2008) on GWs generated by PV anomalies in a horizontal shear with the difference that the wind shear is vertical in our case. A common feature of all of these processes, one which is likely generic for spontaneous-generation phenomena (Vanneste 2008), is that the GW activity is exponentially weak in the limit of small Rossby number, or equivalently large Richardson number. Our study, which is not restricted to this limit, confirms this conclusion; it nevertheless suggests that for reasonable values of the parameters, the GW amplitudes generated by PV anomalies in a vertical shear can be significant—comparable, for instance, with those observed in the low stratosphere by constant-level balloons far from mountain ranges (Hertzog et al. 2008).

In the background flow that we consider, with uniform PV, the small-amplitude PV anomalies are advected passively by the shear. The spectral representation of this dynamics involves a continuous spectrum of singular modes whose phase velocities are in the range of the basic-flow velocity, (−∞, ∞) in the unbounded domain that we assume. The vertical structure of the PV associated with the singular modes is simple: each mode is represented by a Dirac distribution centered at the critical level, where the phase velocity equals the basic flow velocity [see Pedlosky (1979) for a discussion of the analogous continuous spectrum in the quasigeostrophic (QG) Eady problem]. The fields decay rapidly above and below the critical level as far as the inertial levels where the Doppler-shifted frequency is equal to the Coriolis parameter (Jones 1967). Beyond these levels, the structure is oscillatory and can be identified as the GW signature of the (Dirac) PV anomaly.

The first purpose of this paper is to obtain the vertical structure of the singular modes with Dirac PV analytically. The second is to deduce, by integration over the continuous spectrum, the GW response to a vertically smooth, localized PV distribution. The third is to show that this GW response can yield substantial Eliassen–Palm (or pseudomomentum) fluxes at large vertical distances from the PV anomaly. Interestingly, we find that about half (exactly so in the limit of large Richardson number) of the pseudomomentum generated by the advection of PV is transported by GWs over arbitrarily large distances; the other half is deposited in an inertial layer, where substantial wave–flow interactions likely take place.

The plan of the paper is as follows: The general formulation of the problem and its transformation to a dimensionless form are given in section 2. Section 3 is devoted to the derivation of the vertical structure of the singular modes associated with a Dirac in PV. In section 4 we rephrase the result of section 3 in dimensional terms and estimate the amplitude of the pseudomomentum fluxes that can be expected from horizontally monochromatic PV anomalies; these compare with those measured in the lower stratosphere during field campaigns. We also consider PV distributions that are localized horizontally and have a finite depth so that the GW response is transient. In section 5, we summarize our results and discuss their significance for (i) the parameterization of GWs in GCMs, (ii) the transient evolution of baroclinic disturbances, and (iii) the treatment of the more general initial value problem. An appendix is devoted to approximate solutions valid in the limit of large Richardson number.

## 2. General formulation

### a. Disturbance equations and potential vorticity

**u**

_{0}= (Λ

*z*, 0, 0), where Λ > 0 denotes the shear, read as

*u*′,

*υ*′, and

*w*′ are the three components of the velocity disturbance,

*p*′ is the pressure disturbance,

*ρ*is a constant reference density,

_{r}*θ*′ is the potential temperature disturbance,

*θ*is a constant reference potential temperature,

_{r}*g*is the gravity constant,

*f*is the Coriolis parameter, and

*N*

^{2}=

*g*

*θ*

_{0z}/

*θ*is the square of the constant Brunt–Väisälä frequency, with

_{r}*θ*

_{0}(

*y*,

*z*) the background potential temperature. Note that Λ appears in (2.1d) because the background shear flow is in thermal wind balance,

*θ*

_{0y}= −

*θ*Λ/

_{r}f*g*.

*θ*

_{0y}+

*f*

*θ*

_{0z}, being constant. It follows from (2.2) that the PV at any time

*t*is given explicitly in terms of the initial condition

*q*′

_{0}(

*x*,

*z*) =

*q*′(

*x*,

*z*,

*t*= 0) by

### b. Normal-mode decomposition

*q̂*

_{0}is the Fourier transform of

*q*′

_{0}, satisfying

*δ*(

*ξ*) is the Dirac function of the variable

*z*′. The scaling used in (2.8) places the inertial levels of these modes at

*ξ*= ±1 (Inverarity and Shutts 2000).

*ψ*′, related to the perturbation velocity in the (

*x*,

*z*) plane according to

*ψ̂*

_{0}(

*k*,

*z*′) is the amplitude of the normal mode and Ψ(

*ξ*) its vertical structure. Note that this expansion, which describes the part of

*ψ*′ slaved to the PV, is not complete: an additional continuum of singular modes, representing free sheared GWs, would need to be added to the expansion to solve an arbitrary initial value problem.

*u*′,

*υ*′,

*w*′ and the potential temperature

*θ*′ have expansions analogous to (2.10), with

*ψ̂*

_{0}replaced by

*û*

_{0},

*υ̂*

_{0},

*ŵ*

_{0}and

*θ̂*

_{0}, and with Ψ replaced by

*U*,

*V*,

*W*. Introducing these expansions into (2.1a)–(2.1e) and choosing

*ξ*):

## 3. Evaluation of Ψ(*ξ*)

*ξ*> 0 and impose a radiation condition for

*ξ*≫ 1 to obtain a solution that represents an upward-propagating GW. We deduce from this a solution valid for

*ξ*< 0, which represents a downward-propagating GW for

*ξ*≪ −1. The amplitudes of these two solutions are then chosen to satisfy the jump conditions associated with (2.15),

### a. Homogeneous solution for ξ > 0

*η*=

*ξ*

^{2}transforms (2.15) into the canonical form of the hypergeometric equation [Eq. (15.5.1) in Abramowitz and Stegun (1964), hereafter AS]:

*a*+

*b*−

*c*= 0, a relation that is related to the fact that the two inertial levels at

*ξ*= ±1 are logarithmic singularities of (2.15), as described by Jones (1967).

*ξ*> 1 we retain one of the two independent solutions of the hypergeometric equations [see AS, (15.5.8)], written as

*F*denotes the hypergeometric function,

*a*′ =

*a**, and

*b*′ = 1 −

*a*. We retain this solution because its asymptotic form,

It is important to note here that, without the hydrostatic approximation, the equation for Ψ in (2.15) would present turning points in the very far field. At these turning points, the GWs would be entirely reflected back toward the critical levels. This effect, and the fact that we are only interested here in the emission of the GWs by the PV, has motivated our choice to make the hydrostatic approximation.

*ξ*< 1 the solution to (3.2a) is best written as a linear combination of the two independent solutions:

*a*″ = 1 −

*a**,

*b*″ = 1 −

*a*, and

*A*and

*B*are two complex constants [(15.5.3–4) in AS]. To connect this solution to (3.3), we use a transformation formula for

*F*[(15.3.10) in AS] and obtain the asymptotic approximations

*ψ*is the digamma function (see AS, chapter 6). The other coefficients (

_{d}*α*′,

*β*′) and (

*α*″,

*β*″) are defined by the same formulas with (

*a*,

*b*) replaced by (

*a*′,

*b*′) and (

*a*″,

*b*″), respectively.

*ξ*= 1, we follow Booker and Bretherton (1967) and introduce a infinitely small linear damping that shifts the real

*ξ*axis into the lower half of the complex plane so that

*A*and

*B*gives

^{(u)}(

*ξ*).

### b. Solution over the entire domain

*ξ*< 0 can be deduced from Ψ

^{(u)}(

*ξ*) by noting that (2.15) only contains real coefficients and is even. A possible solution is simply

*ξ*→ −∞ since

^{(u)}and Ψ

^{(d)}can be combined to obtain a solution valid in the entire domain that satisfies the jump condition (3.1). This is given by

*ξ*| < 1,

*ξ*= 0 and do not contribute to the jump in (3.1); the second terms do not affect the value of Ψ in

*ξ*= 0 but combine so that Ψ

_{ξ}/

*ξ*

^{2}jumps by 1 at

*ξ*= 0 as required.

*ξ*). First, for small |

*ξ*|, Ψ(

*ξ*) approaches the value

*J*. This asymptotic estimate, derived here using Stirling’s formula for the gamma function [see AS’s (6.1.37)], is also obtained when the quasigeostrophic approximation of (2.15) is solved (see section 1 of the appendix). Second, we obtain from (3.4) and (3.13) that

*ξ*→ −∞ is similar. Accordingly, the amplitude of the GWs in the far field (|

*ξ*| ≫ 1) is

*J*approximation in (3.18) is one of the main results of this paper. Obviously it cannot be recovered in the quasigeostrophic approximation, which filters out GWs completely, but it can be recovered by a WKB treatment of (2.15). This is demonstrated in section 2 of the appendix.

*ξ*). Multiplying (2.15) by

*J*

^{3/2}Ψ* and integrating by parts results in a conservation relation for the nondimensional EP flux

*ξ*= 0, ±1. Using the asymptotic approximation (3.16) shows that

^{ξ}=

*μJ*

^{3/2}|

*E*|

^{2}/2 for |

*ξ*| > 1. The flux

^{ξ}is discontinuous across the inertial level

*ξ*= 1. To evaluate its jump, we use the approximations (3.6a) and (3.6b) valid for

*ξ*→ 1

^{±}to find that

^{ξ}(1

^{+}) −

^{ξ}(1

^{−}) = −

*πJ*

^{3/2}|

*E*|

^{2}|

*α*″|

^{2}. Finally, the Taylor series expansions of (3.14a) and (3.14b) near

*ξ*= 0 shows that the flux is continuous across

*ξ*= 0:

^{ξ}(0

^{+}) −

^{ξ}(0

^{−}) = 0. Using the explicit expression of

*α*″ and Stirling’s formula, our results for the EP flux are summarized as follows:

### c. Results

The four panels in Fig. 1 show the solution (3.13) for Ψ for four values of the Richardson number *J*. For all values of *J*, the streamfunction amplitude decays away from *ξ* = 0 in the region between *ξ* = ±1, where it has an almost constant phase, indicating a nonpropagating character. The decay for small *ξ* is well predicted by the quasigeostrophic approximation Ψ* _{g}* (also shown), which behaves essentially like exp(−

*J*

*ξ*|) (see section 1 of the appendix).

Beyond the inertial levels, that is, for |*ξ*| > 1, the disturbance is propagating, with the real and imaginary part of Ψ in quadrature. The asymptotic behavior *Eξ*^{1/2+iμ} as |*ξ*| → ∞ is also shown in Fig. 1: it corresponds to a pure GW in our context. The amplitude |*E*| of the far-field GW, given in (3.18), is compared in Fig. 2 with its large *J* asymptotic estimate, also given in (3.18). The figure confirms the validity of this estimate and shows that it remains useful for values of *J* as small as 1. The figure also highlights the rapid decrease of the GW amplitude with *J* that is encapsulated in the exponential factor exp(−*J**π*/2) appearing in the asymptotic estimate. A crude argument is suggestive of this dependence: if the quasigeostrophic approximation is used (well beyond its range of validity) up to the inertial levels *ξ* = ±1, the amplitude attained there, and hence the GW amplitude, is predicted to be roughly exp(−*J**π*/2 can be traced to the breakdown of the quasigeostrophic approximation for *ξ* = *O*(1) and the replacement of the decay in exp(−*J**ξ*|) of the quasigeostrophic solution by a decay in exp(−*J*^{−1}*ξ*|). This is demonstrated explicitly by the WKB solution in section 2 of the appendix. Figure 2a also compares the value of Ψ(0) with its large *J* (quasigeostrophic) estimate, confirming the validity and usefulness of the estimate.

The amplitude of the nondimensional EP fluxes, between and beyond the inertial levels *ξ* = ±1, is shown in Fig. 2b. The exact results are compared with the large *J* estimates, which again prove accurate for *J* as small as 1. The figure, like Fig. 2a, illustrates the strong sensitivity of the GW generation to *J*.

## 4. Solutions for various PV distributions

To gauge the possible importance of sheared PV anomalies as a mechanism of GW generation, we now evaluate the GW field produced by a variety of localized anomalies. From now on, we report the results in dimensional form.

We compute the perturbation fields on a grid by evaluating the integral (2.10) numerically for different initial PV distributions *q*′_{0}(*x*, *z*) using the analytical solutions derived in section 3. The computational domain −*X* < *x* < *X* and −*Z* < *z* < *Z* is discretized in a regular grid with 2*L* + 1 points in the horizontal and 2*M* + 1 in the vertical. Correspondingly, the Fourier transform *q̂*_{0}(*k*, *z*) is also discretized and given as *q̂*_{0}(*k _{l}*,

*z*), where

_{m}*k*=

_{l}*πl*/

*X*,

*l*= −

*L*, … ,

*L*and

*z*=

_{m}*πm*/

*Z*,

*m*= −

*M*, … ,

*M*. In the computations presented below, we typically take

*X*= 4000 km,

*Z*= 10 km,

*L*= 2048, and

*M*= 200, which ensures excellent resolutions in both directions of propagation as well as in the spectral space.

*q̂*

_{0}(

*k*,

_{l}*z*), the perturbation PV at different times is computed as

_{m}*x*and

*z*lie on the grid, Δ

*k*=

*π*/(

*LX*) and Δ

*z*=

*Z*/

*M*. The perturbation streamfunction then takes the analogous form

*ψ̂*

_{0}(

*k*,

_{l}*z*) =

_{m}*gρ*

_{r}q̂_{0}(

*k*,

_{l}*z*)/(

_{m}*kθ*Λ

_{r}^{2}) [see (2.10) and (2.14)]. Similar expressions can be written down for

*u*′,

*υ*′,

*w*′, and

*θ*′ using (2.11) and (2.12).

### a. Monochromatic, infinitely thin PV

*q*, in a layer of depth

_{r}*σ*that varies monochromatically with wavelength 2

_{z}*π*/

*k*. Modeling this layer as infinitely thin corresponds to taking

_{r}*ψ*=

_{r}*gρ*/(

_{r}q_{r}*k*Λ

_{r}θ_{r}^{2}). This is of course nothing other than the solution discussed in section 3 except for the dimensional factor. We focus on the dimensional aspect and examine the dimensional EP flux,

*ρ*= 1 PVU, yielding

_{r}q_{r}*ρ*= 10

_{r}q_{r}σ_{z}^{−3}K s

^{−1}. Assuming that this air enters the troposphere at midlatitudes, we take

*ρ*= 1 kg m

_{r}^{−3},

*N*= 0.01 s

^{−1},

*θ*= 300 K, and

_{r}*f*= 10

^{−4}s

^{−1}. For these parameters the dimensional factor, in Eq. (4.5),

^{ξ}needs to be multiplied by 10 Pa to be dimensionally meaningful. This scaling, which is independent of the horizontal wavenumber

*k*, is used in Fig. 2 for the axis on the right of the panel. For

_{r}*J*between 1 and 10, the EP flux is seen to be between 0.1 and 100 mPa. This covers the range of values measured in the lower stratosphere away from mountains during the constant-level-balloon Vorcore campaigns (Hertzog et al. 2008).

### b. Horizontally localized, infinitely thin PV

*σ*gives the width of the PV distribution, by taking in (4.1)

_{x}The vertical velocity field corresponding to this distribution of PV is shown in Fig. 3. The width of the PV anomaly has been taken as *σ _{x}* = 40 km (which corresponds to about 0.5° of longitude when

*f*= 10

^{−4}s

^{−1}); the other parameters are as in the preceding subsection. The four panels are obtained for different values of the shear Λ and hence of

*J*, with the value of

*N*kept constant.

For small values of *J* (*J* = 2 and *J* = 5), there are clear differences in the response to the PV anomaly between a region immediately surrounding the PV anomaly and the two far-field regions. The transition between these three regions can be located around the altitudes *z _{I}* = ±

*fσ*/Λ of the inertial levels of disturbances with wavelength 1/

_{x}*σ*(for

_{x}*J*= 2 and 5,

*z*= ±500 m and

_{I}*z*= ±1 km, respectively). In what follows, we call the transition regions “inertial layers.” Between the inertial layers, the vertical velocity is everywhere positive to the east of the positive PV disturbance (i.e., for

_{I}*x*> 0) and negative to the west. This is because the transverse wind

*υ*′ is cyclonic (with

*υ*′ < 0 for

*x*> 0 and

*υ*′ < 0 for

*x*< 0) and the meridional advection of background potential temperature, −

*f*Λ

*υ*′ in (2.1d), is balanced by the vertical advection

*N*

^{2}

*w*′. Note also that the vertical velocity is almost untilted in the vertical in this region and decreases in amplitude when |

*z*| increases. These indicate that the dynamics near

*z*= 0 is well predicted by the quasigeostrophic theory.

Above and below the inertial layers, the disturbance has a propagating character, with *w*′ changing sign with altitude at a given horizontal location. In these two regions *w*′ is also tilted against the shear, indicating upward propagation in the upper region and downward propagation in the lower one. The GW signal is comparable in magnitude with the signal near the PV anomaly for *J* = 2 (Fig. 3a) but substantially smaller when *J* = 5 (Fig. 3b). For even larger *J* (Figs. 3c and 3d) the GW signal becomes very weak—so weak as to be undetectable for *J* = 25.

*u*′,

*w*′,

*υ*′ and

*θ*′. The EP flux is almost constant between the inertial layers, falls off by a factor of about 2 across the inertial layers, and is constant beyond. This suggests that substantial wave–mean flow interactions can occur at distances of up to a few kilometers from PV anomalies.

### c. Horizontally localized, finite-depth PV

*x*and

*z*at

*t*= 0. Specifically, we choose

The time-dependent streamfunction corresponding to the PV is computed according to (4.2). Because *x* and *t* enter (4.2) only in the combination *x* − Λ*z _{m}t*, it is possible to reduce considerably the computations involved by summing vertical and horizontal translations of the solution in section 4b. To illustrate briefly how the horizontal translations are done, for fixed

*z*, the sum over the index

_{m}*l*of the wavenumber

*k*at some time

_{l}*t*can be inferred from the corresponding sum for

*t*= 0 provided that

*t*is a integer multiple of Δ

*x*/(ΛΔ

*z*). We have adjusted our grid size to take advantage of this for the values of

*t*chosen for the results presented.

Figure 5 shows the evolution of the disturbance PV (gray shading) and of the vertical velocity it produces for *σ _{z}* = 1 km,

*J*= 5, and all other parameters as in the previous sections. The solution is shown only for negative values of

*t*. Indeed, the symmetries in our problem are such that, for the shallow PV disturbances considered, the solutions for positive

*t*are almost symmetric to that at negative

*t*. The background velocity shears the PV, which is strongly tilted against the shear for large negative time and with the shear for large positive time (not shown). Accordingly, the width of the PV distribution is deeply altered, decreasing here by a factor of ∼2 from

*t*= −24 h to

*t*= 0. As a result, the vertical velocity signal has a decreasing width and increasing amplitude as

*t*increases toward 0.

Comparing the four panels in Fig. 5 to the (time independent) disturbance produced by the infinitely thin distribution of Fig. 3b indicates that the GW patterns in the far field are comparable at *t* = −12 h and almost identical at *t* = 0 h. Accordingly, and because the EP flux is a quadratic quantity, it is only in a time interval of a day or so that we can expect the EP flux to approach the values shown in Fig. 4. This last point is confirmed by Fig. 6, which displays the evolution of *F*^{z} evaluated at *z* = +10 km for the finite-depth PV anomaly. For all values of *J*, the EP flux peaks at *t* = 0, with peak values that compare well with the (time independent) values obtained for the infinitely thin distribution [see Fig. 4 and Eq. (4.9)]. The peak in EP flux shown in Fig. 6 broadens as *J* increases because the PV advection also slows. Nevertheless, the characteristic durations of the burst in EP flux remain of the order of a day in the range of *J* considered here.

Although this last result is quite sensitive to the spatial extent of the PV distribution (with the characteristic duration of the EP flux bursts decreasing when the depth of the PV disturbance increases and/or when its width decreases), it clearly illustrates that our mechanism of GWs generation is quite robust: the EP flux in the far field predicted by (4.9) at some time are representative of the flux within a few hours of that time. This is important for the parameterization of the GWs produced by PV anomalies in GCMs since these parameterization schemes can realistically be updated every few hours.

## 5. Summary and applications

### a. Summary

In the presence of a uniform vertical shear, and in the absence of boundaries in the vertical, localized PV anomalies produce disturbances associated with two inertial layers. These layers are located above and below the PV anomaly, at a distance *z _{I}* =

*σ*/Λ, where

_{x}f*σ*is the typical width of the PV anomaly. Between these two inertial layers, the form of the disturbance is qualitatively well predicted by the quasigeostrophic theory, and geostrophic balance is a good approximation for the meridional wind in this region. Accordingly, the decay with altitude of the disturbance amplitude is exponential with a decay rate of the order of

_{x}*N*/(

*fσ*). Beyond the inertial layers, the intrinsic frequency of the disturbance is larger than

_{x}*f*so that the disturbance propagates vertically in the form of a GW. If the disturbance amplitude is substantial at the inertial level, the GW amplitudes produced by this mechanism can be significant. This condition is satisfied provided

*z*/(

_{I}N*fσ*) =

_{x}*N*/Λ =

*J*

*J*

*π*/2)/(2

*J*) for

*J*> 1. Correspondingly, the Eliassen–Palm flux associated with the GW depends also exponentially on

*J*, scaling almost like exp(−

*π*

*J*

*π*

*J*

The robustness of these results has been tested numerically and for the case of a PV disturbance localized horizontally and of finite depth. Horizontal localization spreads the inertial level of the monochromatic case over an inertial layer, and finite depth leads to a time-dependent perturbation. It is nevertheless shown that under these circumstances the GW amplitudes and the EP fluxes in the far field remain well predicted by the monochromatic results, with multiplicative factors that are *O*(1) [see (4.9)]. The characteristic time over which they evolve is also found to exceed a few hours, suggesting that our formula can usefully predict the EP fluxes produced by PV anomalies (if these anomalies are diagnosed every hour, for instance).

### b. Applications

These results can be directly useful for the parameterization of GWs in GCMs that include the middle atmosphere. The dimensional results of section 4 suggest that the EP fluxes in the far field produced by localized PV anomalies compare in amplitude with the EP fluxes measured in the lower stratosphere by almost constant-level balloons during the Vorcore campaign (Hertzog et al. 2008). Although the analytical expressions we give are well suited to this context, notably because the GW parameterization routines typically reevaluate the EP fluxes according to the large-scale flow every hour, they require an estimation of the PV anomalies at subgrid scales.

In most of the parameterizations used currently, the GW EP fluxes from the troposphere toward the middle atmosphere are imposed regardless of the GWs tropospheric sources. There are nevertheless some exceptions, as in Charron and Manzini (2002), where the GW amplitude is larger if fronts are identified. Our formula in (3.20) could well be used in this context since large PV anomalies form during frontogenesis. It could therefore be used to parameterize quantitatively the GW radiated by fronts as well as by other processes that induce localized PV anomalies.

Our results are also relevant to the problem of nonmodal growth of baroclinic disturbances (Farrell 1989). In fact, the inertial levels, which are ignored in balanced approximations, change the boundary conditions at large distances from the critical level from decay conditions into radiation conditions; as a result, the global structure of the solutions in the continuous spectrum is deeply altered. Consequently an inflow PV anomaly of short horizontal scales can have a much larger surface signature than predicted by balanced models. This is because the disturbances produced in this case can have an inertial layer between the PV and the surface. As our solutions are the nongeostrophic counterpart of the building blocks that are used by Bishop and Heifetz (2000) to explain the triggering of storms by upstream PV anomalies or to explain the optimal perturbations evolution by de Vries and Opsteegh (2007), they may be useful to examine these problems in the nongeostrophic case.

Finally, our results could be useful to study the evolution of initial disturbances or of disturbances produced by any external causes (the classical adjustment problem). For this purpose, it should nevertheless be noticed that the modes with Dirac PV are not the only modes associated with the continuous spectrum. In fact, for each value of the phase velocity, there are two further singular modes, with zero PV and singular behavior of the other fields at the inertial levels. These modes are essential to the completeness of the modal representation of the solutions.

## Acknowledgments

This work was supported by the Alliance Programme of the French Foreign Affairs Ministry and British Council. J.V. acknowledges the support of a NERC grant. R.P. acknowledges the support of the ANR project FLOWING.

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## APPENDIX

### Approximate Solutions

#### Quasigeostrophic approximation

*ψ*′

*that can be written as a spectral expansion similar to (2.10), where*

_{g}*ψ̂*

_{0}is as in (2.14) and Ψ(

*ξ*) is replaced by the structure function Ψ

*(*

_{g}*ξ*). This new structure function Ψ

*(*

_{g}*ξ*) is a solution of the QG approximation of the structure equation (2.15), namely

*ξ*| → ∞ is given by

#### WKB approximation

In this section, we solve the differential Eq. (2.15) asymptotically using a WKB method. This method has the interest of providing the solution in terms of (mostly) elementary functions: these reveal the structure of the solution more explicitly than the exact solutions do.

^{(u)}(

*ξ*) valid for

*ξ*> 0, we distinguish four different regions in which Ψ

^{(u)}(

*ξ*) takes a different asymptotic form: (i) an inner region with

*ξ*≪ 1; (ii) an outer region with

*O*(1) =

*ξ*< 1; (iii) an inner region for

*ξ*≈ 1; and (iv) a second outer region for

*ξ*> 1. In the outer regions (ii) and (iv), two independent solutions are found by introducing the WKB ansatz

*A*. Thus, we obtain

*A*

^{(ii)},

*B*

^{(ii)},

*A*

^{(vi)}, and

*B*

^{(vi)}are arbitrary constants. The proper radiation condition as

*ξ*→ ∞ is satisfied by (A.5) provided that

*Z*=

*J*

*z*into (2.15) and obtain at leading order the equation

*ζ*=

*J*(

*ξ*− 1) and obtain the leading-order equation

^{(u)}(

*ξ*) for

*ξ*> 0 with [Ψ

^{(u)}(−

*ξ*)]* for

*ξ*< 0. The jump condition at

*ξ*= 0, applied to (A.6), gives

*x*| → ∞, and −

*π*< arg

*x*< 2

*π*and −2

*π*< arg

*x*<

*π*, respectively [see (9.2.3–4) in AS]. Using this in (A.7) and comparing with the expansion of (A.4) for

*ξ*→ 1

^{−}gives

*ξ*→ 1

^{+}gives

*ξ*implies that

*B*

^{(iv)}= 0, we conclude that

*A*

^{(i)}= 1/(2

*J*

^{3/2}) and, hence, that

*J*

^{3/2}) and that the gravity wave amplitude is |

*A*

^{(iv)}| ∼

*e*

^{−}

^{J}

^{π/2}/(2

*J*), consistent with (3.15) and (3.18), respectively. The EP flux

^{ξ}∼

*e*

^{−}

^{J}

^{π}/8 for |

*ξ*| > 1 follows immediately, consistent with (3.20). The evaluation of the EP flux for |

*ξ*| < 1 is more delicate because it includes terms that are exponentially small in

*J*

Characteristic amplitudes of the disturbances produced by the Dirac PV anomaly: (a) exact and approximate values for the GW amplitude |*E*| given by (3.18) (black solid and gray dashed curves, respectively) and exact and approximate values of Ψ(0) given by (3.15) (gray solid curve and black dots, respectively), (b) exact and approximate values of the EP flux *F ^{ξ}* between the inertial levels (black solid curve and gray solid curve) and exact and approximate values of

*F*beyond the inertial levels (black dashed curve and gray dashed curve); see (3.20). The axis on the right of (b) gives the results in the dimensional form corresponding to the parameter choice (4.6).

^{ξ}Citation: Journal of the Atmospheric Sciences 67, 1; 10.1175/2009JAS3134.1

Characteristic amplitudes of the disturbances produced by the Dirac PV anomaly: (a) exact and approximate values for the GW amplitude |*E*| given by (3.18) (black solid and gray dashed curves, respectively) and exact and approximate values of Ψ(0) given by (3.15) (gray solid curve and black dots, respectively), (b) exact and approximate values of the EP flux *F ^{ξ}* between the inertial levels (black solid curve and gray solid curve) and exact and approximate values of

*F*beyond the inertial levels (black dashed curve and gray dashed curve); see (3.20). The axis on the right of (b) gives the results in the dimensional form corresponding to the parameter choice (4.6).

^{ξ}Citation: Journal of the Atmospheric Sciences 67, 1; 10.1175/2009JAS3134.1

Characteristic amplitudes of the disturbances produced by the Dirac PV anomaly: (a) exact and approximate values for the GW amplitude |*E*| given by (3.18) (black solid and gray dashed curves, respectively) and exact and approximate values of Ψ(0) given by (3.15) (gray solid curve and black dots, respectively), (b) exact and approximate values of the EP flux *F ^{ξ}* between the inertial levels (black solid curve and gray solid curve) and exact and approximate values of

*F*beyond the inertial levels (black dashed curve and gray dashed curve); see (3.20). The axis on the right of (b) gives the results in the dimensional form corresponding to the parameter choice (4.6).

^{ξ}Citation: Journal of the Atmospheric Sciences 67, 1; 10.1175/2009JAS3134.1

Vertical velocity induced by a horizontally localized but infinitely thin PV anomaly [see the PV horizontal profile in (4.7)] for (a) *J* = 2, (b) *J* = 5, (c) *J* = 10, and (d) *J* = 25, with dashed curves indicating negative values. The contour interval, indicated on each panel, varies.

Citation: Journal of the Atmospheric Sciences 67, 1; 10.1175/2009JAS3134.1

Vertical velocity induced by a horizontally localized but infinitely thin PV anomaly [see the PV horizontal profile in (4.7)] for (a) *J* = 2, (b) *J* = 5, (c) *J* = 10, and (d) *J* = 25, with dashed curves indicating negative values. The contour interval, indicated on each panel, varies.

Citation: Journal of the Atmospheric Sciences 67, 1; 10.1175/2009JAS3134.1

Vertical velocity induced by a horizontally localized but infinitely thin PV anomaly [see the PV horizontal profile in (4.7)] for (a) *J* = 2, (b) *J* = 5, (c) *J* = 10, and (d) *J* = 25, with dashed curves indicating negative values. The contour interval, indicated on each panel, varies.

Citation: Journal of the Atmospheric Sciences 67, 1; 10.1175/2009JAS3134.1

Vertical profiles of Eliassen–Palm flux for the four solutions in Fig. 3. Note the rescaling of the flux for the different values of *J*.

Citation: Journal of the Atmospheric Sciences 67, 1; 10.1175/2009JAS3134.1

Vertical profiles of Eliassen–Palm flux for the four solutions in Fig. 3. Note the rescaling of the flux for the different values of *J*.

Citation: Journal of the Atmospheric Sciences 67, 1; 10.1175/2009JAS3134.1

Vertical profiles of Eliassen–Palm flux for the four solutions in Fig. 3. Note the rescaling of the flux for the different values of *J*.

Citation: Journal of the Atmospheric Sciences 67, 1; 10.1175/2009JAS3134.1

Evolution of the vertical velocity field associated with the evolution of a PV disturbance of finite depth *σ _{z}* = 1 km, finite width

*σ*= 40 km, and maximum value of

_{x}*ρ*= 1 PVU, and when the Richardson number

_{r}q_{r}*J*= 5. PV values above 0.1 PVU are shaded; contours for the vertical velocity as in Fig. 3b.

Citation: Journal of the Atmospheric Sciences 67, 1; 10.1175/2009JAS3134.1

Evolution of the vertical velocity field associated with the evolution of a PV disturbance of finite depth *σ _{z}* = 1 km, finite width

*σ*= 40 km, and maximum value of

_{x}*ρ*= 1 PVU, and when the Richardson number

_{r}q_{r}*J*= 5. PV values above 0.1 PVU are shaded; contours for the vertical velocity as in Fig. 3b.

Citation: Journal of the Atmospheric Sciences 67, 1; 10.1175/2009JAS3134.1

Evolution of the vertical velocity field associated with the evolution of a PV disturbance of finite depth *σ _{z}* = 1 km, finite width

*σ*= 40 km, and maximum value of

_{x}*ρ*= 1 PVU, and when the Richardson number

_{r}q_{r}*J*= 5. PV values above 0.1 PVU are shaded; contours for the vertical velocity as in Fig. 3b.

Citation: Journal of the Atmospheric Sciences 67, 1; 10.1175/2009JAS3134.1

Temporal evolution of the far-field EP flux for finite-depth, finite-width PV anomaly and for different values of the Richardson number *J*. Note that the EP flux has been rescaled as in Fig. 4.

Citation: Journal of the Atmospheric Sciences 67, 1; 10.1175/2009JAS3134.1

Temporal evolution of the far-field EP flux for finite-depth, finite-width PV anomaly and for different values of the Richardson number *J*. Note that the EP flux has been rescaled as in Fig. 4.

Citation: Journal of the Atmospheric Sciences 67, 1; 10.1175/2009JAS3134.1

Temporal evolution of the far-field EP flux for finite-depth, finite-width PV anomaly and for different values of the Richardson number *J*. Note that the EP flux has been rescaled as in Fig. 4.

Citation: Journal of the Atmospheric Sciences 67, 1; 10.1175/2009JAS3134.1

^{1}

It is in this context that Scavuzzo et al. (1998) and Lott (2003) attributed the presence of inertia–gravity waves near mountains to a large-scale adjustment to breaking small-scale gravity waves.